Question

Use the following for #5-6 A middle school science teacher wants to conduct some experiments. There are 15 students in the class. The teacher selects the students randomly to work together in groups of five. 5) In how many ways can the teacher combine five of the students for the first group if the order is not important? 6) After the first group of five is selected, in how many ways can the teacher combine five of the remaining students if the order is not important?

Answer 1

• 5) 3003 ways;
• 6) 252 ways.

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5) Use the combination formula:

• C(n, r) = n! / (r!(n-r)!)

In this case, n = 15 (total students) and r = 5 (students in a group).

Substitute and calculate:

• C(15, 5) = 15! / (5!(15-5)!)
• C(15, 5) = 15! / (5!10!)
• C(15, 5) = 3003

The teacher can combine the students in 3003 ways for the first group.

6) After the first group of five is selected, there are 10 students remaining.

Again use the combination formula, with n = 10 and r = 5:

• C(10, 5) = 10! / (5!(10-5)!)
• C(10, 5) = 10! / (5!5!)
• C(10, 5) = 252

The teacher can combine the remaining students in 252 ways for the second group.